1.
The RSA Algorithm
JooSeok Song
2007. 11. 13. Tue
2.
Private-Key Cryptography
traditional private/secret/single key
cryptography uses one key
shared by both sender and receiver
if this key is disclosed communications are
compromised
also is symmetric, parties are equal
hence does not protect sender from receiver
forging a message & claiming is sent by sender
CCLAB
3.
Public-Key Cryptography
probably most significant advance in the 3000
year history of cryptography
uses two keys – a public & a private key
asymmetric since parties are not equal
uses clever application of number theoretic
concepts to function
complements rather than replaces private key
crypto
CCLAB
4.
Public-Key Cryptography
public-key/two-key/asymmetric cryptography
involves the use of two keys:
– a public-key, which may be known by anybody, and
can be used to encrypt messages, and verify
signatures
– a private-key, known only to the recipient, used to
decrypt messages, and sign (create) signatures
is asymmetric because
– those who encrypt messages or verify signatures
cannot decrypt messages or create signatures
CCLAB
6.
Why Public-Key Cryptography?
developed to address two key issues:
– key distribution – how to have secure communications
in general without having to trust a KDC with your key
– digital signatures – how to verify a message comes
intact from the claimed sender
public invention due to Whitfield Diffie & Martin
Hellman at Stanford Uni in 1976
– known earlier in classified community
CCLAB
7.
Public-Key Characteristics
Public-Key algorithms rely on two keys with the
characteristics that it is:
– computationally infeasible to find decryption key
knowing only algorithm & encryption key
– computationally easy to en/decrypt messages when the
relevant (en/decrypt) key is known
– either of the two related keys can be used for
encryption, with the other used for decryption (in some
schemes)
CCLAB
9.
Public-Key Applications
can classify uses into 3 categories:
– encryption/decryption (provide secrecy)
– digital signatures (provide authentication)
– key exchange (of session keys)
some algorithms are suitable for all uses, others
are specific to one
CCLAB
10.
Security of Public Key Schemes
like private key schemes brute force exhaustive
search attack is always theoretically possible
but keys used are too large (>512bits)
security relies on a large enough difference in
difficulty between easy (en/decrypt) and hard
(cryptanalyse) problems
more generally the hard problem is known, its
just made too hard to do in practise
requires the use of very large numbers
hence is slow compared to private key schemes
CCLAB
11.
RSA
by Rivest, Shamir & Adleman of MIT in 1977
best known & widely used public-key scheme
based on exponentiation in a finite (Galois) field
over integers modulo a prime
– nb. exponentiation takes O((log n)3) operations (easy)
uses large integers (eg. 1024 bits)
security due to cost of factoring large numbers
– nb. factorization takes O(e log n log log n) operations (hard)
CCLAB
12.
RSA Key Setup
each user generates a public/private key pair by:
selecting two large primes at random - p, q
computing their system modulus N=p.q
– note ø(N)=(p-1)(q-1)
selecting at random the encryption key e
where 1<e<ø(N), gcd(e,ø(N))=1
solve following equation to find decryption key d
– e.d=1 mod ø(N) and 0≤d≤N
publish their public encryption key: KU={e,N}
keep secret private decryption key: KR={d,p,q}
CCLAB
13.
RSA Use
to encrypt a message M the sender:
– obtains public key of recipient KU={e,N}
– computes: C=Me mod N, where 0≤M<N
to decrypt the ciphertext C the owner:
– uses their private key KR={d,p,q}
– computes: M=Cd mod N
note that the message M must be smaller than
the modulus N (block if needed)
CCLAB
14.
Prime Numbers
prime numbers only have divisors of 1 and self
– they cannot be written as a product of other numbers
– note: 1 is prime, but is generally not of interest
eg. 2,3,5,7 are prime, 4,6,8,9,10 are not
prime numbers are central to number theory
list of prime number less than 200 is:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61
67 71 73 79 83 89 97 101 103 107 109 113 127 131
137 139 149 151 157 163 167 173 179 181 191 193
197 199
CCLAB
15.
Prime Factorisation
to factor a number n is to write it as a product of
other numbers: n=a × b × c
note that factoring a number is relatively hard
compared to multiplying the factors together to
generate the number
the prime factorisation of a number n is when its
written as a product of primes
– eg. 91=7×13 ; 3600=24×32×52
CCLAB
16.
Relatively Prime Numbers & GCD
two numbers a, b are relatively prime if have
no common divisors apart from 1
– eg. 8 & 15 are relatively prime since factors of 8 are
1,2,4,8 and of 15 are 1,3,5,15 and 1 is the only common
factor
conversely can determine the greatest common
divisor by comparing their prime factorizations
and using least powers
– eg. 300=21×31×52 18=21×32 hence
GCD(18,300)=21×31×50=6
CCLAB
17.
Fermat's Theorem
ap-1 mod p = 1
– where p is prime and gcd(a,p)=1
also known as Fermat’s Little Theorem
useful in public key and primality testing
CCLAB
18.
Euler Totient Function ø(n)
when doing arithmetic modulo n
complete set of residues is: 0..n-1
reduced set of residues is those numbers
(residues) which are relatively prime to n
– eg for n=10,
– complete set of residues is {0,1,2,3,4,5,6,7,8,9}
– reduced set of residues is {1,3,7,9}
number of elements in reduced set of residues is
called the Euler Totient Function ø(n)
CCLAB
19.
Euler Totient Function ø(n)
to compute ø(n) need to count number of
elements to be excluded
in general need prime factorization, but
– for p (p prime) ø(p) = p-1
– for p.q (p,q prime)
ø(p.q) = (p-1)(q-1)
eg.
– ø(37) = 36
– ø(21) = (3–1)×(7–1) = 2×6 = 12
CCLAB
20.
Euler's Theorem
a generalisation of Fermat's Theorem
aø(n)mod N = 1
– where gcd(a,N)=1
eg.
–
–
–
–
CCLAB
a=3;n=10;
hence 34 =
a=2;n=11;
hence 210 =
ø(10)=4;
81 = 1 mod 10
ø(11)=10;
1024 = 1 mod 11
21.
Why RSA Works
because of Euler's Theorem:
aø(n)mod N = 1
– where gcd(a,N)=1
in RSA have:
–
–
–
–
N=p.q
ø(N)=(p-1)(q-1)
carefully chosen e & d to be inverses mod ø(N)
hence e.d=1+k.ø(N) for some k
hence :
Cd = (Me)d = M1+k.ø(N) = M1.(Mø(N))q = M1.(1)q =
M1 = M mod N
CCLAB
22.
RSA Example
1.
2.
3.
4.
5.
Select primes: p=17 & q=11
Compute n = pq =17×11=187
Compute ø(n)=(p–1)(q-1)=16×10=160
Select e : gcd(e,160)=1; choose e=7
Determine d: de=1 mod 160 and d < 160
Value is d=23 since 23×7=161= 10×160+1
6. Publish public key KU={7,187}
7. Keep secret private key KR={23,17,11}
CCLAB
23.
RSA Example cont
sample RSA encryption/decryption is:
given message M = 88 (nb. 88<187)
encryption:
C = 887 mod 187 = 11
decryption:
M = 1123 mod 187 = 88
CCLAB
24.
Exponentiation
can use the Square and Multiply Algorithm
a fast, efficient algorithm for exponentiation
concept is based on repeatedly squaring base
and multiplying in the ones that are needed to
compute the result
look at binary representation of exponent
only takes O(log2 n) multiples for number n
– eg. 75 = 74.71 = 3.7 = 10 mod 11
– eg. 3129 = 3128.31 = 5.3 = 4 mod 11
CCLAB
26.
RSA Key Generation
users of RSA must:
– determine two primes at random - p, q
– select either e or d and compute the other
primes p,q must not be easily derived from
modulus N=p.q
– means must be sufficiently large
– typically guess and use probabilistic test
exponents e, d are inverses, so use Inverse
algorithm to compute the other
CCLAB
27.
RSA Security
three approaches to attacking RSA:
– brute force key search (infeasible given size of
numbers)
– mathematical attacks (based on difficulty of computing
ø(N), by factoring modulus N)
– timing attacks (on running of decryption)
CCLAB
28.
Factoring Problem
mathematical approach takes 3 forms:
– factor N=p.q, hence find ø(N) and then d
– determine ø(N) directly and find d
– find d directly
currently believe all equivalent to factoring
– have seen slow improvements over the years
as of Aug-99 best is 130 decimal digits (512) bit with GNFS
– biggest improvement comes from improved algorithm
cf “Quadratic Sieve” to “Generalized Number Field Sieve”
– barring dramatic breakthrough 1024+ bit RSA secure
ensure p, q of similar size and matching other constraints
CCLAB
29.
Timing Attacks
developed in mid-1990’s
exploit timing variations in operations
– eg. multiplying by small vs large number
– or IF's varying which instructions executed
infer operand size based on time taken
RSA exploits time taken in exponentiation
countermeasures
– use constant exponentiation time
– add random delays
– blind values used in calculations
CCLAB
30.
Summary
have considered:
–
–
–
–
–
–
–
CCLAB
prime numbers
Fermat’s and Euler’s Theorems
Primality Testing
Chinese Remainder Theorem
Discrete Logarithms
principles of public-key cryptography
RSA algorithm, implementation, security
31.
Assignments
1. Perform encryption and decryption using RSA
algorithm, as in Figure 1, for the following:
a. p = 3; q = 11, e = 7; M = 5
b. p = 5; q = 11, e = 3; M = 9
Encryption
Plaintext
88
887 mod 187 = 11
Decryption
Ciphertext
11
1123 mod 187 = 88
KU = 7, 187
KR = 23, 187
Figure 1. Example of RSA Algorithm
Plaintext
88
2. In a public-key system using RSA, you intercept
the ciphertext C = 10 sent to a user whose public
key is e = 5, n = 35. What is the plaintext M?
CCLAB
31
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